796 zyxwvutsrqpon IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO. zyx 6, JUNE zy 1987 Wigner Distribution for Finite Duration or Band- Limited Signals and Limiting Cases LEON COHEN Abstiuct-Explicit expressions are given for the evaluation of the Wigner distribution for the case where the signal is of finite duration (or finite bandwidth), and where the functional form of the signal is given by two different expressions. Furthermore, the case where the signal is zero for a time (or frequency band) between the two regions is considered. This general case encompasses many particular cases. We specialize to limiting cases including infinite endpoints. Similar equations are given for the evaluation for the ambiguity function, which is the characteristic function of the Wigner distribution. The catalog- ing of explicit formulas should be of practical utility for the calculation of the Wigner distribution for signals of the type considered. T F(t, zyxwvutsr u) zyxwvutsrq = zyxwvu 1 j’ zyxwvut s*(t - 7)e-2i’ws(t + 7) d7, (1.1) where t and zyxwvutsrq w are the time and angular frequency and zyxwv s (t) is the signal. It can also be written as I. INTRODUCTION HE Wigner distribution [l] has found application in many branches of quantum mechanics and has been applied to various aspects of signal analysis [2]. In the signal theory context it is m a --m S*(w - 7)ef2j7‘S(u + 7) zyxwvuts dr, (1.2) where S( w ) is the Fourier transform of s (t) 4 P-m and in the frequency domain x = w, y = t, R(x) = S(u), +in exponent. (1.6) This allows us to treat both cases simultaneously without repetition of formulas. Because of the particular functional dependence of the integrand on the signal, the evaluation of the Wigner dis- tribution for signals of finite duration (or finite band- width), where the functional form of the signal changes, is quite cumbersome. We present here explicit formulas for such situations. In particular, we consider the general case where -a I x I x1 &(x) x1 I x I x2 &(x) x3 I x I x4 x2 I x I x3 . (1.7) X4IXIOo Rl 1 1 R2 _--________- X1 x2 x3 x4 For finite duration signals, the Wigner distribution is zero for times before the signal starts and after the signal ends, and similarly for the frequency domain. To eval- uate it for thegeneral situation given by (1.7), the follow- ing four inequality conditions must hold separately if the integrand is to have a nonzero value: As the equations we will give will apply to both the fre- x1 I x - 7 I x2 and x1 I x + 7 I x, (1.8a) quency and time domain, it is convenient to mathemati- cally combine both forms into one x1 I x - 7 I x2 and x3 5 x + 7 I x4 (1.8b) ~*(x - 7)ef2”YR(x +- 7) d~ (1.4) - where for the time domain we take x = t, y = w, R(x) = s(t), -in exponent, (1.5) Manuscript received February 17, 1986; revised September 30, 1986. This work was supported in part by a grant from the City University of New York Research Award Program. The author is with the Department of Physics and Astronomy, Hunter College, City University of New York, New York, NY 10021, on leave at the IBM T. J. Watson Research Center, Yorktown Heights, NY 10598. IEEE Log Number 8613864. x3 5 x - 7 I x4 and x1 I x + 7 I x2 (1.8~) x3 I x - T I x4 and x3 I x + 7 I x4. (1.8d) Solving each for the variable x gives the conditions and values for the integral. The conditions are determined by the different ordering of the mean times or frequencies defined by ‘We emphasize, however, that when the signal is zero, the Wigner dis- tribution is not necessarily so. Similarly, if the spectrum does not include certain frequencies, it does not follow that the Wigner distribution is zero for those frequencies. This is obvious from the equations that follow and is one of the important properties of the Wigner distribution which will be discussed in detail elsewhere. 0096-3518/87/0600-0796$01.00 zyxw 0 1987 IEEE